A378561 Number of ways to place k nonattacking anassas on an n X n chess board. Triangle T(n,k) read by rows.

1, 1, 1, 1, 4, 3, 1, 9, 22, 14, 1, 16, 82, 156, 90, 1, 25, 220, 840, 1366, 738, 1, 36, 485, 3100, 9796, 14288, 7364, 1, 49, 938, 9030, 46816, 129360, 174112, 86608, 1, 64, 1652, 22344, 172116, 767424, 1916776, 2424880, 1173240, 1, 81, 2712, 49056, 525756, 3442740, 13682320, 31572720, 38019496, 17990600

Offset

0,5

Comments

Anassas (also called semi-rook+semi-bishop) are chess pieces with 2 moves: one horizontal or vertical and one diagonal.

Links

Table of n, a(n) for n=0..54.

S. Chaiken, C. R. H. Hanusa, and T. Zaslavsky, A q-Queens Problem. V. Some of Our Favorite Pieces: Queens, Bishops, Rooks, and Nightriders, J. Korean Math. Soc., 57(6): 1407-1433, 2020; see also arXiv preprint, arXiv:1609.00853 [math.CO], 2016-2020.

Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 716-722.

Eder G. Santos, Counting non-attacking chess pieces placements: Bishops and Anassas, arXiv:2411.16492 [math.CO], 2024.

Formula

T(n,k) = Sum_{j=0..ceiling(k/2)} j! * binomial(n-k+j,j) * Stirling2(n,n-k+j) * 2^(k-2*j) * (binomial(k-j,j-1) + binomial(k-j+1,j)).

Example

Triangle begins:
  1;
  1  1;
  1  4   3;
  1  9  22   14;
  1 16  82  156   90;
  1 25 220  840 1366   738;
  1 36 485 3100 9796 14288 7364;
  ...

Prog

(SageMath) print([sum([factorial(j)*binomial(n-k+j, j)*stirling_number2(n, n-k+j)*2^(k-2*j)*(binomial(k-j, j-1)+binomial(k-j+1, j)) for j in [0..ceil(k/2)]]) for n in [0..10] for k in [0..n]])

Crossrefs

Columns k=0-1 give: A000012, A000290.

Main diagonal gives A088789(n+1).

Sequence in context: A193011 A214859 A123160 * A109692 A039758 A157894

Adjacent sequences: A378558 A378559 A378560 * A378562 A378563 A378564

Keyword

nonn,easy,tabl

Author

Eder G. Santos, Nov 30 2024

Status

approved